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PLOTTING SCALAR AND VECTOR FIELDS^19
~ Electric field lines Equipotentials
Figure 2.1 Equipotentials and Electric Field Lines of Two Parallel Line Charges
Usually we want to construct surfaces of constant CP, which in this case are cylinders
nested around the line charges. Because there is symmetry in the z direction, these
surfaces can be plotted in two dimensions as the dashed, circular contours shown in
Figure 2.1. The centers of these circles are located along the x-axis from 1 < x < 03
for positive values of <P, and from --M < x < - 1 for negative values of CP. CP = 0
lies on the y-axis, which you can think of as a circle of infinite radius with its center at infinity. If the contours have evenly spaced constant values of <P, the regions of highest line density show where the function is most rapidly varying with position.
2.1.2 Plotting Vector Fields
Because vectors have both magnitude and direction, plots for their fields are usually much more complicatedthan for scalar fields. For example, the Cartesian components of the electric field of the preceding example can be calculated to be
A vector field is typically field vector at every point in
(2.2)
"[ I
x2 - y2 - 1 7 E 0 [(x^ -^ 1)2^ +^ y2][(x^ +^ 1)2^ +^ y2]
(2.3)
I.
"[ 2 XY
7rE0 [(x^ -^ 1)2^ +^ y2][(x^ +^ 1)2^ +^ y2]
drawn by constructing lines which are tangent to the space. By convention, the^ density of these field lines
20 DIFFERENTIAL AND INTEGRAL OPERATIONS
indicates the magnitude of the field, while arrows show its direction. If we suppose
an electric field line for Equations 2.2 and 2.3 is given by the equation y = y ( x ) , then
With some work, Equation 2.4 can be integrated to give
x2 + 0 , - c)2 = 1 + 2, (^) (2.5)
where c is the constant of integration. This constant can be varied between --03 and -03 to generate the entire family of field lines. For this case, these lines are circles centered on the y-axis at y = c with radii given by I/=. They are shown as the solid lines in Figure 2.1. The arrows indicate how the field points from the positive to the negative charge. Notice the lines are most densely packed directly between the charges where the electric field is strongest.
2.2 INTEGRALOPERATORS
2.2.1 Integral Operator Notation
The gradient, divergence, and curl operations, which we will review later in this chapter, are naturally in operator form. That is, they can be represented by a symbol that operates on another quantity. For example, the gradient of @ is written as v@. Here the operator is v, which acts on the operand @ to give us the gradient. In contrast, integral operations are comonly not written in operator form. The integral of f ( x ) over x is often expressed as
which is not in operator form because the integral and the operand f(x) are inter- mingled. We can, however, put Equation 2.6 in operator form by reorganizing the terms in this equation:
Now the operator d x acts on f ( x ) to form the integral, just as the v operator acts on @ to form the gradient. In practice, the integral operator is moved to the right, passing through all the terms of the integrand that do not depend on the integration variable. For example,
/ d x x2(x + y ) y 2 = y 2 / d x x2(x + y ). J J
